Related papers: Quantum graphs -- Generic eigenfunctions and their…
The notion of network connectivity is used to characterize the robustness and failure tolerance of networks, with high connectivity being a desirable feature. In this paper, we develop a novel approach to the problem of identifying critical…
We develop a unified quantum framework for subgraph counting in graphs. We encode a graph on $N$ vertices into a quantum state on $2\lceil \log_2 N \rceil$ working qubits and $2$ ancilla qubits using its adjacency list, with worst-case gate…
The objective of this note is to provide an interpretation of the discrete version of Morse inequalities, following Witten's approach via supersymmetric quantum mechanics, adapted to finite graphs, as a particular instance of Morse-Witten…
In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order…
We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard…
In this work we introduce a concept of complexity for undirected graphs in terms of the spectral analysis of the Laplacian operator defined by the incidence matrix of the graph. Precisely, we compute the norm of the vector of eigenvalues of…
We prove upper and lower bounds for the number of zeroes of linear combinations of Schr\"odinger eigenfunctions on metric (quantum) graphs. These bounds are distinct from both the interval and manifolds. We complement these bounds by giving…
We study the nodal count of the so-called bi-dendral graphs and show that it exhibits an anomaly: the nodal surplus is never equal to 0 or $\beta$, the first Betti number of the graph. According to the nodal-magnetic theorem, this means…
This paper introduces an efficient quantum computing method for reducing special graphs in the context of the graph coloring problem. The special graphs considered include both symmetric and non-symmetric graphs where the axis passes…
This is an introduction to graph theory, from a geometric and analytic viewpoint. A finite graph $X$ is described by its adjacency matrix $d\in M_N(0,1)$, which can be thought of as being a kind of discrete Laplacian, and we first discuss…
We formulate and prove a Gelfand-Levitan trace formula for general quantum graphs with arbitrary edge lengths and coupling conditions which cover all self-adjoint operators on quantum graphs, except for a set of measure zero. The formula is…
Quantum graphs have recently emerged as models of nonlinear optical, quantum confined systems with exquisite topological sensitivity and the potential for predicting structures with an intrinsic, off-resonance response approaching the…
We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $\nu_n$ is proportional…
The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral…
We demonstrate that quantum correlations can emerge from the statistical correlations of random discrete models, without an a priori assumption that the random models are quantum mechanical in nature, that is without considering…
Our goal is to efficiently compute low-dimensional latent coordinates for nodes in an input graph -- known as graph embedding -- for subsequent data processing such as clustering. Focusing on finite graphs that are interpreted as uniform…
We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as $\delta$ conditions, on some or all of the graph vertices. We classify the eigenvalue asymptotics as…
A chain of quantum subgroups of the quantum automorphism group of finite graphs has been introduced. It generalizes the construction of J. Bichon (see [3]) in a sense. A better bound of the non zero eigenvalues of the graph Laplacian has…
In this paper we show that all nodes can be found optimally for almost all random Erd\H{o}s-R\'enyi ${\mathcal G}(n,p)$ graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices,…
The study of very large graphs is a prominent theme in modern-day mathematics. In this paper we develop a rigorous foundation for studying the space of finite labelled graphs and their limits. These limiting objects are naturally countable…